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[Maxima-commits] CVS: maxima/doc/info lbfgs.texi,1.7,1.8
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From: Robert D. <rob...@us...> - 2009-07-12 18:20:54
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Update of /cvsroot/maxima/maxima/doc/info In directory 23jxhf1.ch3.sourceforge.com:/tmp/cvs-serv9759/doc/info Modified Files: lbfgs.texi Log Message: In lbfgs, mention optional argument for gradient. Index: lbfgs.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/lbfgs.texi,v retrieving revision 1.7 retrieving revision 1.8 diff -u -d -r1.7 -r1.8 --- lbfgs.texi 28 Nov 2007 03:36:50 -0000 1.7 +++ lbfgs.texi 12 Jul 2009 18:20:48 -0000 1.8 @@ -31,11 +31,16 @@ @section Functions and Variables for lbfgs @deffn {Function} lbfgs (@var{FOM}, @var{X}, @var{X0}, @var{epsilon}, @var{iprint}) +@deffnx {Function} lbfgs ([@var{FOM}, @var{grad}] @var{X}, @var{X0}, @var{epsilon}, @var{iprint}) Finds an approximate solution of the unconstrained minimization of the figure of merit @var{FOM} over the list of variables @var{X}, starting from initial estimates @var{X0}, -such that @math{norm grad FOM < epsilon max(1, norm X)}. +such that @math{norm(grad(FOM)) < epsilon*max(1, norm(X))}. + +@var{grad}, if present, is the gradient of @var{FOM} with respect to the variables @var{X}. +@var{grad} is a list, with one element for each element of @var{X}. +If not present, the gradient is computed automatically by symbolic differentiation. The algorithm applied is a limited-memory quasi-Newton (BFGS) algorithm [1]. It is called a limited-memory method because a low-rank approximation of the @@ -338,6 +343,80 @@ (%o8) @end example +Gradient of FOM is specified (instead of computing it automatically). + +@c ===beg=== +@c load (lbfgs)$ +@c F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6; +@c F_grad : map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]); +@c estimates : lbfgs ([F(a, b, c), F_grad], [a, b, c], [0, 0, 0], 1e-4, [1, 0]); +@c ===end=== +@example +(%i1) load (lbfgs)$ +(%i2) F(a, b, c) := (a - 5)^2 + (b - 3)^4 + (c - 2)^6; + 2 4 6 +(%o2) F(a, b, c) := (a - 5) + (b - 3) + (c - 2) +(%i3) F_grad : map (lambda ([x], diff (F(a, b, c), x)), [a, b, c]); + 3 5 +(%o3) [2 (a - 5), 4 (b - 3) , 6 (c - 2) ] +(%i4) estimates : lbfgs ([F(a, b, c), F_grad], [a, b, c], [0, 0, 0], 1e-4, [1, 0]); +************************************************* + N= 3 NUMBER OF CORRECTIONS=25 + INITIAL VALUES + F= 1.700000000000000D+02 GNORM= 2.205175729958953D+02 +************************************************* + +@end example +@ifnottex +@example + I NFN FUNC GNORM STEPLENGTH + + 1 2 6.632967565917638D+01 6.498411132518770D+01 4.534785987412505D-03 + 2 3 4.368890936228036D+01 3.784147651974131D+01 1.000000000000000D+00 + 3 4 2.685298972775190D+01 1.640262125898521D+01 1.000000000000000D+00 + 4 5 1.909064767659852D+01 9.733664001790506D+00 1.000000000000000D+00 + 5 6 1.006493272061515D+01 6.344808151880209D+00 1.000000000000000D+00 + 6 7 1.215263596054294D+00 2.204727876126879D+00 1.000000000000000D+00 + 7 8 1.080252896385334D-02 1.431637116951849D-01 1.000000000000000D+00 + 8 9 8.407195124830908D-03 1.126344579730013D-01 1.000000000000000D+00 + 9 10 5.022091686198527D-03 7.750731829225274D-02 1.000000000000000D+00 + 10 11 2.277152808939775D-03 5.032810859286795D-02 1.000000000000000D+00 + 11 12 6.489384688303218D-04 1.932007150271008D-02 1.000000000000000D+00 + 12 13 2.075791943844548D-04 6.964319310814364D-03 1.000000000000000D+00 + 13 14 7.349472666162257D-05 4.017449067849554D-03 1.000000000000000D+00 + 14 15 2.293617477985237D-05 1.334590390856715D-03 1.000000000000000D+00 + 15 16 7.683645404048675D-06 6.011057038099201D-04 1.000000000000000D+00 +@end example +@end ifnottex +@tex +\halign{\hfil\tt#&\quad\hfil\tt#\quad&\tt#\hfil\quad&\tt#\hfil\quad&\tt#\hfil\cr +I& NFN& FUNC& GNORM& STEPLENGTH\cr +&&&&\cr + 1& 2& 6.632967565917638D+01& 6.498411132518770D+01& 4.534785987412505D-03\cr + 2& 3& 4.368890936228036D+01& 3.784147651974131D+01& 1.000000000000000D+00\cr + 3& 4& 2.685298972775190D+01& 1.640262125898521D+01& 1.000000000000000D+00\cr + 4& 5& 1.909064767659852D+01& 9.733664001790506D+00& 1.000000000000000D+00\cr + 5& 6& 1.006493272061515D+01& 6.344808151880209D+00& 1.000000000000000D+00\cr + 6& 7& 1.215263596054294D+00& 2.204727876126879D+00& 1.000000000000000D+00\cr + 7& 8& 1.080252896385334D-02& 1.431637116951849D-01& 1.000000000000000D+00\cr + 8& 9& 8.407195124830908D-03& 1.126344579730013D-01& 1.000000000000000D+00\cr + 9& 10& 5.022091686198527D-03& 7.750731829225274D-02& 1.000000000000000D+00\cr + 10& 11& 2.277152808939775D-03& 5.032810859286795D-02& 1.000000000000000D+00\cr + 11& 12& 6.489384688303218D-04& 1.932007150271008D-02& 1.000000000000000D+00\cr + 12& 13& 2.075791943844548D-04& 6.964319310814364D-03& 1.000000000000000D+00\cr + 13& 14& 7.349472666162257D-05& 4.017449067849554D-03& 1.000000000000000D+00\cr + 14& 15& 2.293617477985237D-05& 1.334590390856715D-03& 1.000000000000000D+00\cr + 15& 16& 7.683645404048675D-06& 6.011057038099201D-04& 1.000000000000000D+00\cr +} +@end tex +@example + + THE MINIMIZATION TERMINATED WITHOUT DETECTING ERRORS. + IFLAG = 0 +(%o4) [a = 5.000086823042934, b = 3.05239542970518, + c = 1.927980629919583] +@end example + @opencatbox @category{Package lbfgs} @closecatbox |